Finite Group Databasebeta

← #29733653

Subgroups — #29733653

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840149C211yesmaximal, characteristic, complemented
2C2×#16840119C211yesmaximal, characteristic, complemented
3#21451576C211yesmaximal, characteristic, complemented
4#21446822C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446789C211yesmaximal, characteristic, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446167C212yesmaximal, complemented
9#21446153C212yesmaximal, complemented
10#21445585C211yesmaximal, characteristic, complemented
11C2×C61×#540C211yesmaximal, characteristic, complemented
12C22×#11758200C2212yes—
13C22×#11758199C2212yes—
14C2×#14519765C2211yescharacteristic
15C2×#14519756C2211yescharacteristic
16C2×#14519751C2211yescharacteristic
17#16840149—24no—
18#16840119—24no—
19C2×#14519700C2212yes—
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×#14519697C2212yes—
23C2×C22⋊#92418C2212yes—
24C2×#14519688C2211yescharacteristic
25C2×C4×#92418C2211yescharacteristic
26#16839684—22no—
27#16839678—22no—
28#16839668C2211yescharacteristic
29#16839664C2211yescharacteristic
30#16839632C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C61×#540—24no—
35C61×#540—24no—
36C2×#127×C61C2211yescharacteristic
37C2×#125×C61C2211yescharacteristic
38C2×#123×C61—22no—
39C2×#121×C61—22no—
40C2×#121×C61C2212yes—
41C2×C4×D4×C61C2212yes—
42C2×#117×C61C2211yescharacteristic
43C22×Dic122C2311yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—21no—
46C2×#11758200—24no—
47C2×#11758200—24no—
48C2×#11758200—42no—
49C2×#11758199—24no—
50C2×#11758199—24no—
51C2×#11758199—42no—
52C22×#92418C2312yes—
53C22×#92418C2312yes—
54#14519765—22no—
55#14519765—22no—
56#14519756—42no—
57#14519751—42no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66#14519700—24no—
67#14519698—42no—
68#14519698—42no—
69#14519697—24no—
70C22⋊#92418—24no—
71#14519688—22no—
72C4×#92418—22no—
73#14519684—22no—
74C22×D4×C61—22no—
75C22×D4×C61—22no—
76C22×D4×C61C2312yes—
77C23×C4×C61—22no—
78C23×C4×C61C2311yescharacteristic
79#127×C61—22no—
80#127×C61D414yescomplemented
81#125×C61—22no—
82#125×C61D414yescomplemented
83#123×C61—216no—
84#121×C61—216no—
85#121×C61—24no—
86#121×C61—24no—
87C4×D4×C61—24no—
88C4×D4×C61—24no—
89#117×C61—22no—
90#117×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61D412yes—
93C2×C4⋊C4×C61C2312yes—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61D412yes—
97C2×C42×C61C2311yescharacteristic
98C2×C42×C61C2311yescharacteristic
99C24×C61—22no—
100C24×C61C2×D411yescharacteristic
101C23×D61—21no—
102C2×D4×C61—28no—
103C2×D4×C61—28no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—82no—
110C2×D4×C61—82no—
111C2×D4×C61—82no—
112C2×D4×C61—82no—
113C2×D4×C61C2×D414yes—
114C22×C4×C61—22no—
115C22×C4×C61—28no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—82no—
121C22×C4×C61—82no—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—42no—
127C2×C22⋊D61—82no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—22no—
134C2×D244—22no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C4⋊C4×C61—28no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61C2×D414yes—
143C22⋊C4×C61—28no—
144C22⋊C4×C61—28no—
145C22⋊C4×C61—24no—
146C42×C61—21no—
147C42×C61—21no—
148C42×C61C2×D412yes—
149C42×C61C2×D412yes—
150C22⋊Dic61—42no—
151#11758200—48no—
152#11758199—48no—
153#11758198—42no—
154#11758198—44no—
155#11758198—82no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—82no—
159#11758196—42no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418#4114yes—
166C23×C61—21no—
167C23×C61—24no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61C22×D411yescharacteristic
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—42no—
183C22×D61—42no—
184D4×C61—24no—
185D4×C61—24no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—88no—
189D4×C61—88no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—24no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—88no—
204C2×C4×C61#12014yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4112yes—
208C2×C4×C61H(5,𝔽2)12yes—
209C22⋊D61—84no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217D244—42no—
218D244—42no—
219Dic122—42no—
220Dic122—42no—
221#92418—24no—
222#92418—42no—
223#25611—611nomaximal, Sylow
224C22×C61—42no—
225C22×C61—41no—
226C22×C61—44no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61—44no—
230C22×C61—42no—
231C22×C61—44no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61C2×#12011yescharacteristic
237C22×C61#53211yescharacteristic
238C22×C61#53511yescharacteristic
239C22×C61#57612yes—
240C22×C61#57712yes—
241C2×D61—82no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C2×D61—82no—
246C4×C61—24no—
247C4×C61—41no—
248C4×C61—42no—
249C4×C61—44no—
250C4×C61#54612yes—
251C4×C61#55012yes—
252Dic61—81no—
253Dic61—81no—
254Dic61—81no—
255C2×#540D6111yescharacteristic, complemented
256C2×#496—611no—
257C2×#488—611no—
258#2620—611no—
259#2000—612no—
260#1999—612no—
261#1992—611no—
262#1957—611no—
263#1879—612no—
264#1877—612no—
265#1750—611no—
266C2×C61—81no—
267C2×C61—84no—
268C2×C61#243111yescharacteristic
269C2×C61#301212yes—
270C2×C61#317912yes—
271C2×C61#322011yescharacteristic
272C2×C61#322111yescharacteristic
273D61—161no—
274C22×SD16—612no—
275C22×D8—612no—
276#540—24no—
277#540—24no—
278C2×#127C2×D6111yescharacteristic
279C2×#125C2×D6111yescharacteristic
280C2×#124—611no—
281C2×#123—22no—
282C2×C22⋊Q8—611no—
283C2×#121—22no—
284C2×#121—611no—
285C2×#121C2×D6112yes—
286C2×C4×D4C2×D6112yes—
287C2×#117C2×D6111yescharacteristic
288#496—1224no—
289#488—1224no—
290C2×#103—612no—
291C2×#102—612no—
292C2×#102—612no—
293C2×#102—612no—
294C2×C22⋊C8—612no—
295C2×#97—611no—
296C2×C4×C8—611no—
297#396—1222no—
298#394—1222no—
299#391—611no—
300#390—611no—
301#384—611no—
302#340—612no—
303#340—612no—
304#336—612no—
305C61#2561111yescharacteristic, complemented, Sylow
306C22×Q8—611no—
307C22×D4—22no—
308C22×D4—22no—
309C22×D4—611no—
310C22×D4—1221no—
311C22×D4C22×D6112yes—
312C23×C4—22no—
313C23×C4C22×D6111yescharacteristic
314C2×SD16—1224no—
315C2×SD16—1224no—
316C2×SD16—2442no—
317C2×D8—1224no—
318C2×D8—1224no—
319C2×D8—2442no—
320C22×C8—612no—
321C22×C8—612no—
322#127—22no—
323#127C22⋊D6114yescomplemented
324#125—22no—
325#125C22⋊D6114yescomplemented
326#124—1222no—
327#124—1222no—
328#123—216no—
329C22⋊Q8—2442no—
330#121—216no—
331#121—24no—
332#121—24no—
333#121—2442no—
334C4×D4—24no—
335C4×D4—24no—
336#117—22no—
337#117—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4C22⋊D6112yes—
344C2×C4⋊C4C22×D6112yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1222no—
351C2×C22⋊C4C22⋊D6112yes—
352C2×C42C22×D6111yescharacteristic
353C2×C42C22×D6111yescharacteristic
354#103—1224no—
355#102—1224no—
356#102—2442no—
357#102—2442no—
358C22⋊C8—1224no—
359#97—1222no—
360C4×C8—1222no—
361#95—1222no—
362C24—22no—
363C24—1221no—
364C24C2×C22⋊D6111yescharacteristic
365C2×Q8—1221no—
366C2×Q8—1221no—
367C2×Q8—1222no—
368C2×Q8—1222no—
369C2×D4—28no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—28no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—1222no—
383C2×D4—1222no—
384C2×D4—4882no—
385C2×D4—2442no—
386C2×D4C2×C22⋊D6114yes—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4C2×C22⋊D6112yes—
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6112yes—
401C22×C4C23×D6111yescharacteristic
402SD16—2448no—
403D8—2448no—
404C2×C8—614no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C4⋊C4—22no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—4882no—
416C4⋊C4—2442no—
417C4⋊C4C2×C22⋊D6114yes—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—24no—
421C22⋊C4—4882no—
422C22⋊C4—2442no—
423C22⋊C4—2442no—
424C22⋊C4—2444no—
425C42—21no—
426C42—21no—
427C42C2×C22⋊D6112yes—
428C42C2×C22⋊D6112yes—
429C23—21no—
430C23—24no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23—21no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—2441no—
442C23—2441no—
443C23—2442no—
444C23—2442no—
445C23—2441no—
446C23C22×C22⋊D6111yescharacteristic
447Q8—2442no—
448Q8—2442no—
449D4—24no—
450D4—24no—
451D4—42no—
452D4—42no—
453D4—88no—
454D4—88no—
455D4—4884no—
456D4—2442no—
457D4—2442no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—88no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4#1451974614yes—
480C2×C4C22×C22⋊D6111yescharacteristic
481C2×C4C22×C22⋊D6111yescharacteristic
482C2×C4#41×D6112yes—
483C2×C4#1451982312yes—
484C8—1224no—
485C8—2442no—
486C22—44no—
487C22—42no—
488C22—44no—
489C22—42no—
490C22—41no—
491C22—44no—
492C22—41no—
493C22—41no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—4882no—
499C22—4882no—
500C22—4881no—
501C22—4881no—
502C22—4881no—
503C22C2×#1451974611yescharacteristic
504C22#1684030811yescharacteristic
505C22#1684032611yescharacteristic
506C22#1684056512yes—
507C22#1684057312yes—
508C4—24no—
509C4—42no—
510C4—44no—
511C4—41no—
512C4—4881no—
513C4—4881no—
514C4—4881no—
515C4#1684040012yes—
516C4#1684043112yes—
517C2—81no—
518C2—84no—
519C2—9761no—
520C2#2144943411yescharacteristic
521C2#2145512312yes—
522C2#2145650412yes—
523C2#2145686311yescharacteristic
524C2#2145687411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#540C2
socleC23×C61C22×D4
hypercenterC2×#540D61
hypocenterC61#25611
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840149C2semidirect
2. C2×#16840119C2semidirect
3. #21451576C2semidirect
4. #21446822C2semidirect
5. #21446815C2semidirect
6. #21446789C2semidirect
7. #21446626C2semidirect
8. #21446167C2semidirect
9. #21446153C2semidirect
10. #21445585C2semidirect
11. C2×C61×#540C2semidirect
80. #127×C61D4semidirect
82. #125×C61D4semidirect
255. C2×#540D61semidirect
305. C61#25611semidirect
323. #127C22⋊D61semidirect
325. #125C22⋊D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168401492
C2×#168401192
#214515762
#214468222
#214468152
#214467892
#214466262
#214461672
#214461532
#214455852
C2×C61×#5402
#2561161